Textbook Question
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(a) Describe the motion of the object over the interval [0,6].
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.2.77a
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
โซโโด (4๐โ ๐ยฒ) d๐
Verified step by step guidance1
Step 1: Understand the problem. A midpoint Riemann sum is a method to approximate the value of a definite integral by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed. The goal is to express this sum in sigma notation for an arbitrary number of subintervals, n.
Step 2: Define the interval and subintervals. The integral โซโโด (4๐ - ๐ยฒ) d๐ is over the interval [0, 4]. Divide this interval into n subintervals of equal width, ฮ๐ = (4 - 0)/n = 4/n.
Step 3: Determine the midpoints of the subintervals. The midpoints of the subintervals are given by ๐แตข = a + (i - 0.5)ฮ๐, where a = 0 is the starting point of the interval, i is the index of the subinterval (ranging from 1 to n), and ฮ๐ = 4/n.
Step 4: Write the function value at the midpoints. For each midpoint ๐แตข, evaluate the function f(๐) = 4๐ - ๐ยฒ. Substitute ๐แตข into the function to get f(๐แตข) = 4(0 + (i - 0.5)(4/n)) - (0 + (i - 0.5)(4/n))ยฒ.
Step 5: Write the midpoint Riemann sum in sigma notation. The sum is given by Sโ = ฮฃแตขโโโฟ f(๐แตข)ฮ๐, where ฮ๐ = 4/n and f(๐แตข) is the function value at the midpoint. Substitute ฮ๐ and f(๐แตข) into the formula to express the sum in terms of n and i.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Riemann Sum
A Midpoint Riemann Sum is a method for approximating the value of a definite integral. It involves dividing the interval into 'n' subintervals, calculating the midpoint of each subinterval, and then evaluating the function at these midpoints. The sum of these function values, multiplied by the width of the subintervals, provides an estimate of the area under the curve.
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Left, Right, & Midpoint Riemann Sums
Sigma Notation
Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (ฮฃ) to indicate summation, along with an index of summation that specifies the starting and ending values. In the context of Riemann sums, sigma notation is used to express the sum of function values at midpoints across all subintervals.
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04:22Sigma Notation
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as โซ from 'a' to 'b' of f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be approximated using Riemann sums, which provide a numerical method to estimate the area when the exact integral is difficult to compute.
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Related Practice
Textbook Question
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(a) Write the left and right Riemann sums in sigma notation for an arbitrary value of n.
โซโยน cos โปยน ๐ d๐
Textbook Question
Area functions for the same linear function Let ฦ(t) = 2t โ 2 and consider the two area functions A (๐) = โซโหฃ ฦ(t) dt and F(๐) = โซโหฃ ฦ(t) dt .
(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (๐) , for ๐ โฅ 1 .
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Textbook Question
Properties of integrals Use only the fact that โซโโด 3๐ (4 โ๐) d๐ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(a) โซโโฐ 3๐(4 โ ๐) d(๐)
Textbook Question
Substitutions Suppose ฦ is an even function with โซโโธ ฦ(๐) d๐ = 9 . Evaluate each integral.
(a) โซยนโโ ๐ฦ(๐ยฒ) d๐
Textbook Question
Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the ๐-direction and 2b in the y-direction is (๐ยฒ/aยฒ) + (yยฒ /bยฒ) = 1.
(a) Let dยฒ denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ โa, a] to show that the average value of dยฒ is (aยฒ + 2bยฒ) /3 .